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Prove that Linear bounded automata LBA ⊂ PSPACE in TOC?
Linear Bounded Automaton (LBA) is a restricted form of Turing Machine in which input tape is finite.
Example
Prove that LBA ⊂ PSPACE
PSPACE is a superset of the set of context-sensitive languages.
Now to prove LBA=PSPACE,
We use theorem of Space compression with tape reduction which states that,
For every k-tape S(n) space-bounded off-line Turing Machine M and constant c>0,there exists a one-tape cS(n) space-bounded off-line turing machine N such that L(M)=L(N).
Following identity holds for −
DSPACE(S(n))=DSPACE(O(S(n)))
and NSPACE(S(n))=NSPACE(O(S(n)))
Since LBA is one-tape n space-bounded Turing Machine it follows −
LBA=NSPACE(n)---------------------(1)
Now by Savitch theorem, if S is fully space constructible and S(n)>log(n) then
NSPACE(S(n)) ⊆DSPACE(S^{2}(n)) -------------(2)
Final proof
LBA=NSPACE(n)............by(1)
⊆DSPACE(n^{2})............by(2)
⊂DSPACE(n^{3})............by Space Hierarchy Theorem
⊆PSPACE
Space Hierarchy then requires S(n) which is fully space-constructible.